Quantum Mechanics: Matching Concepts with Wavefunctions

This question requires matching physical concepts (List-I) with their corresponding mathematical wavefunctions ($\psi$) (List-II). We will analyze each item based on established quantum mechanics principles and the provided correct answer.

List-I Concepts and List-II Wavefunctions

Here are the items presented in List-I and List-II:

• List-I: A. Sommerfield, B. Bloch, C. Hydrogen, D. One Dimensional potential box
• List-II: I. $\psi_n(r) = \frac{1}{\sqrt{\pi a^3}} e^{-r/a}$, II. $\psi_{nk}(r) = e^{ik\cdot r} u_{nk}(r)$, III. $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$, IV. $\psi_n(r) = \frac{e^{ik\cdot r}}{V^{1/2}}$

Explanation of Matches

The correct matching is determined by the characteristic forms of the wavefunctions associated with each concept:

• A - Sommerfield: Matches with IV. The wavefunction $\psi_n(r) = \frac{e^{ik\cdot r}}{V^{1/2}}$ represents a plane wave, often used as a basis state in models considered within the scope of atomic structure theories, including extensions like those by Sommerfeld.
• B - Bloch: Matches with II. The wavefunction $\psi_{nk}(r) = e^{ik\cdot r} u_{nk}(r)$ is the defining form of a Bloch wave, crucial for describing electrons in periodic potentials found in solids.
• C - Hydrogen: Matches with I. The wavefunction $\psi_n(r) = \frac{1}{\sqrt{\pi a^3}} e^{-r/a}}$ is the ground state ($n=1$) radial wavefunction for the Hydrogen atom, expressed in spherical coordinates.
• D - One Dimensional potential box: Matches with III. The wavefunction $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$ is the exact solution for the energy eigenfunctions of a particle confined within a one-dimensional infinite potential well of length L.

Correct Matching Summary

Based on the analysis, the correct pairing is:

This corresponds to the option A-IV, B-II, C-I, D-III.